Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, find the first term given two terms from an arithmetic sequence. Find the first term or \(a_{1}\) of an arithmetic sequence if \(a_{7}=21\) and \(a_{15}=42\).

Step-by-Step Solution

Answer

The first term of the sequence is 5.25.

Step 1: Understand the Arithmetic Sequence Formula

In an arithmetic sequence, each term increases by a common difference. The general formula for the nth term of an arithmetic sequence is given by: \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference.

Step 2: Write Equations for Given Terms

Given two terms: \(a_7 = 21\) and \(a_{15} = 42\). Substitute these into the formula: \(21 = a_1 + 6d\) and \(42 = a_1 + 14d\).

Step 3: Subtract the Equations to Solve for d

Subtract the first equation from the second to eliminate \(a_1\). This gives us: \((a_1 + 14d) - (a_1 + 6d) = 42 - 21\), which simplifies to \(8d = 21\). Solve for \(d\) to get \(d = \frac{21}{8}\).

Step 4: Substitute d Back to Find a_1

Take the expression for \(d = \frac{21}{8}\) and substitute back into one of the original equations. Use \(21 = a_1 + 6\left(\frac{21}{8}\right)\). Simplify this to find \(a_1\):- \(21 = a_1 + \frac{126}{8}\)- Convert \(\frac{126}{8}\) to \(15.75\) - Thus, \(a_1 = 21 - 15.75 = 5.25\).

Step 5: Verify the Solution

To ensure the solution is correct, check if using \(a_1 = 5.25\) and \(d = \frac{21}{8}\) gives the correct values for both terms:- For \(a_7\): \(a_7 = 5.25 + 6\left(\frac{21}{8}\right) = 21\)- For \(a_{15}\): \(a_{15} = 5.25 + 14\left(\frac{21}{8}\right) = 42\)- Both calculations match the given terms, confirming the solution is correct.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key factor. It's the constant amount that you add (or subtract) to move from one term to the next in the sequence. The common difference is typically denoted by the symbol \(d\).

• If \(d\) is positive, the sequence increases.
• If \(d\) is negative, the sequence decreases.

To find the common difference when you know two terms in an arithmetic sequence, you would generally use the formula: \( d = \frac{a_m - a_n}{m - n} \), where \(a_n\) and \(a_m\) are the known terms in the sequence, and \(n\) and \(m\) are their respective term numbers. In the example provided, having found \(d = \frac{21}{8}\) from the terms \(a_7 = 21\) and \(a_{15} = 42\) is crucial for working out the sequence.

Nth Term

The \(n\)th term in an arithmetic sequence is a formula used to find any term in the sequence without writing out all previous terms. It's given by the formula:\[a_n = a_1 + (n-1) \cdot d\]Here, \(a_n\) represents the \(n\)th term you want to find, \(a_1\) is the first term, \(n\) is the position of the term, and \(d\) is the common difference. Using this formula, you can quickly calculate the value of any term if you know the common difference and the first term. For instance, from the problem we were discussing, if you are asked to find the 7th term and you know \(a_1 = 5.25\) and \(d = \frac{21}{8}\), you'd substitute into the equation to find that \( a_7 = 21\), as confirmed in the solution.

First Term

The first term of an arithmetic sequence, represented as \(a_1\), is the starting point of the sequence. It's the term used in conjunction with the common difference to calculate other terms in the sequence. In many problems, you might be tasked with finding \(a_1\) given other terms in the sequence. For example, using the equations derived from other terms, such as \( 21 = a_1 + 6d\) for our particular problem, you rearrange to solve for \(a_1\). Calculating further gives \( a_1 = 5.25\), which means every other term in this sequence will be calculated starting from this value. Therefore, knowing \(a_1\) provides the foundation for understanding and deriving the rest of the sequence.

Sequence Formula

The sequence formula for an arithmetic sequence is the blueprint for calculating each term. The formula, \( a_n = a_1 + (n-1)d \), allows you to find any \(n\)th term without enumerating previous terms in the sequence explicitly.

• \(a_n\) is the term you calculate.
• \(a_1\) is the first term.
• \(d\) is the common difference.

Applying this formula is essential for efficiency and accuracy, especially when dealing with large sequences. In solving the exercise, substituting the known \(a_1\) and \(d\) into this formula made it possible to verify the solution for terms \(a_7\) and \(a_{15}\). Thus, the sequence formula is indispensable for both constructing and understanding arithmetic sequences.